Verified answer

Ответ:

a) [tex]\displaystyle y'=3\;ln7\cdot\;7^{3x+2}[/tex]

b) [tex]\displaystyle y'=-5\;ln4\cdot\;4^{-5x+2}[/tex]

c) [tex]\displaystyle y'=3\;ln4\cdot\;4^{x+2}[/tex]

d) [tex]\displaystyle y'=-3\;ln10\cdot\;10^{3x-4}[/tex]

e) [tex]\displaystyle y'=\frac{3^{\sqrt{t}}\;ln3 }{\sqrt{t} }[/tex]

f) [tex]\displaystyle y'=\frac{5^{\sqrt{t-2}}\;ln2 }{2\sqrt{t-2} }[/tex]

Объяснение:

Найти производную функции.

Формулы:

[tex]\boxed {\displaystyle \bf (a^u)'=a^u\cdot ln\;a\cdot u'}[/tex]     [tex]\boxed {\displaystyle \bf (x^n)'=nx^{n-1}}[/tex]

[tex]\displaystyle \bf a) \;y=7^{3x+2}[/tex]

[tex]\displaystyle y'=7^{3x+2}ln7\cdot (3x+2)'=7^{3x+2}ln\;7\cdot 3=3\;ln7\cdot\;7^{3x+2}[/tex]

[tex]\displaystyle \bf b) \;y=4^{-5x+2}[/tex]

[tex]\displaystyle y'=4^{-5x+2}ln4\cdot (-5x+2)'=4^{-5x+2}ln\;4\cdot (-5)=-5\;ln4\cdot\;4^{-5x+2}[/tex]

[tex]\displaystyle \bf c) \;y=3\cdot 4^{x+2}[/tex]

[tex]\displaystyle y'=3\cdot 4^{x+2}ln4\cdot (x+2)'=3\cdot 4^{x+2}ln\;4\cdot 1=3\;ln4\cdot\;4^{x+2}[/tex]

[tex]\displaystyle \bf d) \;y=-10^{3x-2}[/tex]

[tex]\displaystyle y'=(-1)\cdot 10^{3x-4}ln10\cdot (3x-4)'=(-1)\cdot 10^{3x-4}ln\;10\cdot 3=-3\;ln10\cdot\;10^{3x-4}[/tex]

[tex]\displaystyle \bf e) \;s=2\cdot 3^{\sqrt{t} }=2\cdot 3^{t^{\frac{1}{2} }}[/tex]

[tex]\displaystyle y'=2\cdot 3^{t^{\frac{1}{2} }}ln3\cdot (t^{\frac{1}{2} })'=2\cdot 3^{t^{\frac{1}{2} }}ln3\cdot \frac{1}{2}t^{-\frac{1}{2} } =\frac{3^{\sqrt{t}}\;ln3 }{\sqrt{t} }[/tex]

[tex]\displaystyle \bf f) \;s=5\cdot 2^{\sqrt{t-2} }=5\cdot 2^{(t-2)^{\frac{1}{2} }[/tex]

[tex]\displaystyle y'=5\cdot 2^{(t-2)^{\frac{1}{2} }}ln2\cdot ((t-2)^{\frac{1}{2} })'=5\cdot 2^{(t-2)^{\frac{1}{2} }}ln2\cdot \frac{1}{2}(t-2)^{-\frac{1}{2} } =\frac{5^{\sqrt{t-2}}\;ln2 }{2\sqrt{t-2} }[/tex]